List of trigonometric identities

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In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Pythagorean identities[edit]

The basic relationship between the sine and cosine is given by the Pythagorean identity:

where means and means

This can be viewed as a version of the Pythagorean theorem, and follows from the equation for the unit circle. This equation can be solved for either the sine or the cosine:

where the sign depends on the quadrant of

Dividing this identity by , , or both yields the following identities:

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

Each trigonometric function in terms of each of the other five.[1]
in terms of

Reflections, shifts, and periodicity[edit]

By examining the unit circle, one can establish the following properties of the trigonometric functions.

Reflections[edit]

Unit circle with a swept angle theta plotted at coordinates (a,b). As the angle is reflected in increments of one-quarter pi (45 degrees), the coordinates are transformed. For a transformation of one-quarter pi (45 degrees, or 90 - theta), the coordinates are transformed to (b,a). Another increment of the angle of reflection by one-quarter pi (90 degrees total, or 180 - theta) transforms the coordinates to (-a,b). A third increment of the angle of reflection by another one-quarter pi (135 degrees total, or 270 - theta) transforms the coordinates to (-b,-a). A final increment of one-quarter pi (180 degrees total, or 360 - theta) transforms the coordinates to (a,-b).
Transformation of coordinates (a,b) when shifting the reflection angle in increments of .

When the direction of a Euclidean vector is represented by an angle this is the angle determined by the free vector (starting at the origin) and the positive -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive -axis. If a line (vector) with direction is reflected about a line with direction then the direction angle of this reflected line (vector) has the value

The values of the trigonometric functions of these angles for specific angles satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.[2]

reflected in [3]
odd/even identities
reflected in reflected in reflected in reflected in
compare to

Shifts and periodicity[edit]

Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).
Transformation of coordinates (a,b) when shifting the angle in increments of .
Shift by one quarter period Shift by one half period Shift by full periods[4] Period

Angle sum and difference identities[edit]

Illustration of angle addition formulae for the sine and cosine of acute angles. Emphasized segment is of unit length.

These are also known as the angle addition and subtraction theorems (or formulae).

These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.

Sine [5][6]
Cosine [6][7]
Tangent [6][8]
Cosecant [9]
Secant [9]
Cotangent [6][10]
Arcsine [11]
Arccosine [12]
Arctangent [13]
Arccotangent

Sines and cosines of sums of infinitely many angles[edit]

When the series converges absolutely then

Because the series converges absolutely, it is necessarily the case that and In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.

Tangents and cotangents of sums[edit]

Let (for ) be the kth-degree elementary symmetric polynomial in the variables

for that is,

Then

using the sine and cosine sum formulae above.

The number of terms on the right side depends on the number of terms on the left side.

For example:

and so on. The case of only finitely many terms can be proved by mathematical induction.[14]

Secants and cosecants of sums[edit]

where is the kth-degree elementary symmetric polynomial in the n variables and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[15] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

Multiple-angle formulae[edit]

Tn is the nth Chebyshev polynomial [16]
de Moivre's formula, i is the imaginary unit [17]

Multiple-angle formulae[edit]

Double-angle formulae[edit]

Formulae for twice an angle.[18]

Triple-angle formulae[edit]

Formulae for triple angles.[18]

Multiple-angle and half-angle formulae[edit]

[citation needed]
[citation needed]

Chebyshev method[edit]

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the (n − 1)th and (n − 2)th values.[19]

cos(nx) can be computed from cos((n − 1)x), cos((n − 2)x), and cos(x) with

cos(nx) = 2 · cos x · cos((n − 1)x) − cos((n − 2)x).

This can be proved by adding together the formulae

cos((n − 1)x + x) = cos((n − 1)x) cos x − sin((n − 1)x) sin x
cos((n − 1)xx) = cos((n − 1)x) cos x + sin((n − 1)x) sin x.

It follows by induction that cos(nx) is a polynomial of the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

Similarly, sin(nx) can be computed from sin((n − 1)x), sin((n − 2)x), and cos(x) with

sin(nx) = 2 · cos x · sin((n − 1)x) − sin((n − 2)x).

This can be proved by adding formulae for sin((n − 1)x + x) and sin((n − 1)xx).

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

Half-angle formulae[edit]

[20][21]

Also

Table[edit]

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

Sine Cosine Tangent Cotangent
Double-angle formulae[22][23]
Triple-angle formulae[16][24]
Half-angle formulae[20][21]

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.[citation needed]

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Power-reduction formulae[edit]

Obtained by solving the second and third versions of the cosine double-angle formula.

Sine Cosine Other

and in general terms of powers of or the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem[citation needed].

Cosine Sine

Product-to-sum and sum-to-product identities[edit]

The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.

Product-to-sum[25]
Sum-to-product[26]

Hermite's cotangent identity[edit]

Charles Hermite demonstrated the following identity.[27] Suppose are complex numbers, no two of which differ by an integer multiple of π. Let

(in particular, being an empty product, is 1). Then

The simplest non-trivial example is the case n = 2:

Ptolemy's theorem[edit]

Ptolemy's theorem can be expressed in the language of modern trigonometry as:

If w + x + y + z = π, then:

(The first three equalities are trivial rearrangements; the fourth is the substance of this identity.)

Finite products of trigonometric functions[edit]

For coprime integers n, m

where Tn is the Chebyshev polynomial.

The following relationship holds for the sine function

More generally [28]

Linear combinations[edit]

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of and .

Sine and cosine[edit]

The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[29][30]

where and are defined as so:

given that

Arbitrary phase shift[edit]

More generally, for arbitrary phase shifts, we have

where and satisfy:

More than two sinusoids[edit]

The general case reads[30]

where

and

Lagrange's trigonometric identities[edit]

These identities, named after Joseph Louis Lagrange, are:[31][32][33]

for

A related function is the following function of called the Dirichlet kernel.

Certain linear fractional transformations[edit]

If is given by the linear fractional transformation

and similarly

then

More tersely stated, if for all we let be what we called above, then

If is the slope of a line, then is the slope of its rotation through an angle of

Relation to the complex exponential function[edit]

Euler's formula states that, for any real number x:

,[34]

where i is the imaginary unit. Substituting -x for x gives us:

.

These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,

[35]
[36]

These formulae are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = e e means that

cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).

That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.

The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm.

Function Inverse function[37]

Infinite product formulae[edit]

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[38][39]

Inverse trigonometric functions[edit]

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[40]

Taking the multiplicative inverse of both sides of the each equation above results in the equations for The right hand side of the formula above will always be flipped. For example, the equation for is:

while the equations for and are:

The following identities are implied by the reflection identities. They hold whenever are in the domains of the relevant functions.

Also,[41]

Identities without variables[edit]

In terms of the arctangent function we have[41]

The curious identity known as Morrie's law,

is a special case of an identity that contains one variable:

Similarly,

is a special case of an identity with  = 20°:

For the case  = 15°,

For the case  = 10°,

The same cosine identity is

Similarly,

Similarly,

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

Other cosine identities include:[42]

and so forth for all odd numbers, and hence

Many of those curious identities stem from more general facts like the following:

[43]

and

Combining these gives us

If n is an odd number () we can make use of the symmetries to get

The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:

Computing π[edit]

An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula:

or, alternatively, by using an identity of Leonhard Euler:

or by using Pythagorean triples:

Others include:

[44][41]
[44]
[41]

Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = Σn−1
k=1
arctan tk ∈ (π/4, 3π/4)
, let tn = tan(π/2 − θn) = cot θn. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1). In particular, the computed tn will be rational whenever all the t1, ..., tn−1 values are rational. With these values,

where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the tk values is not within (−1, 1). Note that if t = p/q is rational, then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2p2, q2 + p2).

For example, for n = 3 terms,

for any a, b, c, d > 0.

An identity of Euclid[edit]

Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:

Ptolemy used this proposition to compute some angles in his table of chords.

Composition of trigonometric functions[edit]

These identities involve a trigonometric function of a trigonometric function:[45]

where Ji are Bessel functions.

Further "conditional" identities for the case α + β + γ = 180°[edit]

The following formulae apply to arbitrary plane triangles and follow from as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).

Historical shorthands[edit]

The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.

Miscellaneous[edit]

Dirichlet kernel[edit]

The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:

The convolution of any integrable function of period with the Dirichlet kernel coincides with the function's th-degree Fourier approximation. The same holds for any measure or generalized function.

Tangent half-angle substitution[edit]

If we set

then[46]

where sometimes abbreviated to cis x.

When this substitution of for tan x/2 is used in calculus, it follows that is replaced by 2t/1 + t2, is replaced by 1 − t2/1 + t2 and the differential dx is replaced by 2 dt/1 + t2. Thereby one converts rational functions of and to rational functions of in order to find their antiderivatives.

Viète's infinite product[edit]

See also[edit]

Notes[edit]

  1. ^ Abramowitz and Stegun, p. 73, 4.3.45
  2. ^ Selby 1970, p. 188
  3. ^ Abramowitz and Stegun, p. 72, 4.3.13–15
  4. ^ Abramowitz and Stegun, p. 72, 4.3.7–9
  5. ^ Abramowitz and Stegun, p. 72, 4.3.16
  6. ^ a b c d Weisstein, Eric W. "Trigonometric Addition Formulas". MathWorld.
  7. ^ Abramowitz and Stegun, p. 72, 4.3.17
  8. ^ Abramowitz and Stegun, p. 72, 4.3.18
  9. ^ a b "Angle Sum and Difference Identities". www.milefoot.com. Retrieved 2019-10-12.
  10. ^ Abramowitz and Stegun, p. 72, 4.3.19
  11. ^ Abramowitz and Stegun, p. 80, 4.4.32
  12. ^ Abramowitz and Stegun, p. 80, 4.4.33
  13. ^ Abramowitz and Stegun, p. 80, 4.4.34
  14. ^ Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC '89 (Portland US-OR, 1989-07). New York: ACM. pp. 207–211. doi:10.1145/74540.74566. ISBN 0-89791-325-6.
  15. ^ Michael Hardy (August–September 2016). "On Tangents and Secants of Infinite Sums". American Mathematical Monthly. 123 (7): 701–703. doi:10.4169/amer.math.monthly.123.7.701. S2CID 126310545.
  16. ^ a b Weisstein, Eric W. "Multiple-Angle Formulas". MathWorld.
  17. ^ Abramowitz and Stegun, p. 74, 4.3.48
  18. ^ a b Selby 1970, pg. 190
  19. ^ Ward, Ken. "Multiple angles recursive formula". Ken Ward's Mathematics Pages.
  20. ^ a b Abramowitz and Stegun, p. 72, 4.3.20–22
  21. ^ a b Weisstein, Eric W. "Half-Angle Formulas". MathWorld.
  22. ^ Abramowitz and Stegun, p. 72, 4.3.24–26
  23. ^ Weisstein, Eric W. "Double-Angle Formulas". MathWorld.
  24. ^ Abramowitz and Stegun, p. 72, 4.3.27–28
  25. ^ Abramowitz and Stegun, p. 72, 4.3.31–33
  26. ^ Abramowitz and Stegun, p. 72, 4.3.34–39
  27. ^ Johnson, Warren P. (Apr 2010). "Trigonometric Identities à la Hermite". American Mathematical Monthly. 117 (4): 311–327. doi:10.4169/000298910x480784. S2CID 29690311.
  28. ^ "Product Identity Multiple Angle".
  29. ^ Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.
  30. ^ a b Weisstein, Eric W. "Harmonic Addition Theorem". MathWorld.
  31. ^ Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". American Journal of Physics. 21 (2): 140. Bibcode:1953AmJPh..21..140M. doi:10.1119/1.1933371.
  32. ^ Agarwal, Ravi P.; O'Regan, Donal (2008). Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems (illustrated ed.). Springer Science & Business Media. p. 185. ISBN 9780387791463. Extract of page 185
  33. ^ Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". Handbook of Mathematical Formulas and Integrals (4th ed.). Academic Press. ISBN 978-0-12-374288-9.
  34. ^ Abramowitz and Stegun, p. 74, 4.3.47
  35. ^ Abramowitz and Stegun, p. 71, 4.3.2
  36. ^ Abramowitz and Stegun, p. 71, 4.3.1
  37. ^ Abramowitz and Stegun, p. 80, 4.4.26–31
  38. ^ Abramowitz and Stegun, p. 75, 4.3.89–90
  39. ^ Abramowitz and Stegun, p. 85, 4.5.68–69
  40. ^ Abramowitz & Stegun 1972, p. 73, 4.3.45
  41. ^ a b c d Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
  42. ^ Humble, Steve (Nov 2004). "Grandma's identity". Mathematical Gazette. 88: 524–525. doi:10.1017/s0025557200176223. S2CID 125105552.
  43. ^ Weisstein, Eric W. "Sine". MathWorld.
  44. ^ a b Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, Proofs Without Words (1993, Mathematical Association of America), p. 39.
  45. ^ Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972, formulae 9.1.42–9.1.45
  46. ^ Abramowitz and Stegun, p. 72, 4.3.23

References[edit]

External links[edit]